The 2018 Mixed Integer Programming workshop took place at Clemson University (Greenville, South Carolina) June 18–21. The goal of the MIP workshop series is to discuss recent research in Mixed Integer Programming, and the 2018 edition consisted of a single track of 20 invited talks, together with a poster session with 26 posters. The six papers published in this special volume are related to talks presented at the workshop. All the papers underwent a rigorous peer-review process; we are very grateful to the anonymous referees, and to the guest editors who handled the submissions for this special issue (Claudia D’Ambrosio, Marcos Goycoolea, Oktay Gunluk, Jim Luedtke, Michele Monaci, Jean-Philippe Richard). We are also grateful to the local organization committee (Akshay Gupte, Matthew Saltzman, Cole Smith) for making this workshop possible.
On behalf of the program committee (Philipp Christophel, Simge Küçükyavuz, Ruth Misener, Giacomo Nannicini, Alejandro Toriello).
Daniel Bienstock and Giacomo Nannicini, New York, May 2020.

The family of critical node detection problems asks for finding a subset of vertices, deletion of which minimizes or maximizes a predefined connectivity measure on the remaining network. We study a problem of this family called the k-vertex cut problem. The problem asks for determining the minimum weight subset of nodes whose removal disconnects a graph into at least k components. We provide two new integer linear programming formulations, along with families of strengthening valid inequalities. Both models involve an exponential number of constraints for which we provide poly-time separation procedures and design the respective branch-and-cut algorithms. In the first formulation one representative vertex is chosen for each of the k mutually disconnected vertex subsets of the remaining graph. In the second formulation, the model is derived from the perspective of a two-phase Stackelberg game in which a leader deletes the vertices in the first phase, and in the second phase a follower builds connected components in the remaining graph. Our computational study demonstrates that a hybrid model in which valid inequalities of both formulations are combined significantly outperforms the state-of-the-art exact methods from the literature.

We consider global optimization of nonconvex problems whose factorable reformulations contain a collection of multilinear equations of the form ze=∏v∈ezv, e∈E, where E denotes a set of subsets of cardinality at least two of a ground set. Important special cases include multilinear and polynomial optimization problems. The multilinear polytope is the convex hull of the set of binary points z satisfying the system of multilinear equations given above. Recently Del Pia and Khajavirad introduced running intersection inequalities, a family of facet-defining inequalities for the multilinear polytope. In this paper we address the separation problem for this class of inequalities. We first prove that separating flower inequalities, a subclass of running intersection inequalities, is NP-hard. Subsequently, for multilinear polytopes of fixed degree, we devise an efficient polynomial-time algorithm for separating running intersection inequalities and embed the proposed cutting-plane generation scheme at every node of the branch-and-reduce global solver BARON. To evaluate the effectiveness of the proposed method we consider two test sets: randomly generated multilinear and polynomial optimization problems of degree three and four, and computer vision instances from an image restoration problem Results show that running intersection cuts significantly improve the performance of BARON and lead to an average CPU time reduction of 50% for the random test set and of 63% for the image restoration test set.

We study two-stage robust optimization problems with mixed discrete-continuous decisions in both stages. Despite their broad range of applications, these problems pose two fundamental challenges: (i) they constitute infinite-dimensional problems that require a finite-dimensional approximation, and (ii) the presence of discrete recourse decisions typically prohibits duality-based solution schemes. We address the first challenge by studying a K-adaptability formulation that selects K candidate recourse policies before observing the realization of the uncertain parameters and that implements the best of these policies after the realization is known. We address the second challenge through a branch-and-bound scheme that enjoys asymptotic convergence in general and finite convergence under specific conditions. We illustrate the performance of our algorithm in numerical experiments involving benchmark data from several application domains.

We present a novel formulation for startup cost computation in the unit commitment problem (UC). Both our proposed formulation and existing formulations in the literature are placed in a formal, theoretical dominance hierarchy based on their respective linear programming relaxations. Our proposed formulation is tested empirically against existing formulations on large-scale UC instances drawn from real-world data. While requiring more variables than the current state-of-the-art formulation, our proposed formulation requires fewer constraints, and is empirically demonstrated to be as tight as a perfect formulation for startup costs. This tightening can reduce the computational burden in comparison to existing formulations, especially for UC instances with large reserve margins and high penetration levels of renewables.

A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with K∗cuts derived from conic certificates for continuous primal-dual conic subproblems. Under the assumption that all subproblems are well-posed, the algorithm detects infeasibility or unboundedness or returns an optimal solution in finite time. Using properties of the conic certificates, we show that the K∗ cuts imply certain practically-relevant guarantees about the quality of the polyhedral relaxations, and demonstrate how to maintain helpful guarantees when the LP solver uses a positive feasibility tolerance. We discuss how to disaggregateK∗ cuts in order to tighten the polyhedral relaxations and thereby improve the speed of convergence, and propose fast heuristic methods of obtaining useful K∗ cuts. Our new open source MI-conic solver Pajarito (github.com/JuliaOpt/Pajarito.jl) uses an external mixed-integer linear solver to manage the search tree and an external continuous conic solver for subproblems. Benchmarking on a library of mixed-integer second-order cone (MISOCP) problems, we find that Pajarito greatly outperforms Bonmin (the leading open source alternative) and is competitive with CPLEX’s specialized MISOCP algorithm. We demonstrate the robustness of Pajarito by solving diverse MI-conic problems involving mixtures of positive semidefinite, second-order, and exponential cones, and provide evidence for the practical value of our analyses and enhancements of K∗ cuts.

Split cuts are arguably the most effective class of cutting planes within a branch-and-cut framework for solving general Mixed-Integer Programs (MIP). Sparsity, on the other hand, is a common characteristic of MIP problems, and it is an important part of why the simplex method works so well inside branch-and-cut. In this work, we evaluate the strength of split cuts that exploit sparsity. In particular, we show that restricting ourselves to sparse disjunctions—and furthermore, ones that have small disjunctive coefficients—still leads to a significant portion of the total gap closed with arbitrary split cuts. We also show how to exploit sparsity structure that is implicit in the MIP formulation to produce splits that are sparse yet still effective. Our results indicate that one possibility to produce good split cuts is to try and exploit such structure.